Every battery case I've seen (for phones with non-removable batteries) occupies the port. Meaning that you can't use it for other interfaces. So that seems like an unlikely solution (not to say it isn't possible or doesn't exist). I realize this doesn't exactly match the criteria presented by the question, but please bear with me. What if you used a USB C Multiport Adapter? Along with a Male USB 3.0 to Female USB 3.1 adapter. Apple's verison seems the most reputable (all the other manufacturers selling something similar on Amazon are companies I've never heard of). If it works with USB C on a MacBook, then why shouldn't it work on a phone? All USB ports must meet a certain standard.
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In Conclusion, You Could Do This:
Connect phone to multiport adapter.
Connected to the multiport adapter is a USB 3.0 to USB C adapter (for connecting to your drone controller)
Connected to the USB C on the multiport adapter is a phone charger of somesort.
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Can someone verify that this does work? It seems like it would work to me, considering the fact that the USB interface is standardized. Thanks!